A new fuzzy robust control strategy for the nonlinear supply chain system in the presence of lead times is proposed. Based on Takagi-Sugeno fuzzy control system, the fuzzy control model of the nonlinear supply chain system with lead times is constructed. Additionally, we design a fuzzy robust H∞ control strategy taking the definition of maximal overlapped-rules group into consideration to restrain the impacts such as those caused by lead times, switching actions among submodels, and customers’ stochastic demands. This control strategy can not only guarantee that the nonlinear supply chain system is robustly asymptotically stable but also realize soft switching among subsystems of the nonlinear supply chain to make the less fluctuation of the system variables by introducing the membership function of fuzzy system. The comparisons between the proposed fuzzy robust H∞ control strategy and the robust H∞ control strategy are finally illustrated through numerical simulations on a two-stage nonlinear supply chain with lead times.
National Social Science Foundation of China17BGL2221. Introduction
Over the recent years, a large number of companies realize the value-added importance of supply chain (SC) system and have cooperated as a part of it [1]. Efficient management of distribution, production, and supply in the SC has critical influence on business success [2]. However, SC system is more sensitive to the existence of lead time. Lead time, which is affected by the physical distance between the seller and the buyer, transportation mode, manufacturer’s production capability, and technology in practice [3], can result in oscillation and instability of the SC system directly. Therefore, effectively restraining the impact of lead time on the SC system can be one of the major challenging issues to be resolved for the node companies in competition [4].
For the controllable lead time, Mahajan and Venugopal [5] studied the impacts of the reduction of lead time on the retailer and manufacturer’s costs. For a two-stage SC consisting of a manufacturer and a retailer, Leng and Parlar [6] investigated game-theoretic models of the reduction of lead time. According to the reduction of lead time caused by the added crashing cost, Li et al. [7] studied the coordination problem of a decentralized SC. Glock [8] proposed alternative approaches on the reduction of lead time and their impacts on the safety inventory and the expected total cost of the integrated inventory system. Further, a model of the divergent SC to study how to minimize the expected total cost and reduce lead times to find the optimal production, inventory, and routing decisions has been described by Jha and Shanker [9]. With the help of lead time variation control, Heydari [10] developed an incentive scheme to realize the service level coordination in a two-stage SC.
On the other hand, for the uncontrollable lead time, Garcia et al. [11] proposed an Internal Model Control (IMC) approach to control the production inventory in a SC with lead times. By utilizing the multimodel scheme, the IMC control approach can realize the online identification of lead times. In addition, Xu and Rong [12] utilized the minimum variance control theory to derive the order-up-to policy for the SC with time-varying lead time. Taleizadeh et al. [13] performed a particle swarm optimization to access the inventory problem of the chance-constraint joint single vendor-single buyer with changeable lead time. To the aim of restraining the bullwhip effect of uncertain SCs with vendor order placement lead time, a robust optimization strategy has been highlighted by Li and Liu [14]. Garcia et al. [15] incorporated an IMC scheme in production inventory control system of a complete SC to online identify lead times. Further, Han et al. [16] analyzed the approximate optimal inventory control problem of SC networks with lead time and proposed a suboptimal inventory replenishment strategy to effectively reduce bullwhip effect and improve the performance of SC networks system. Movahed and Zhang [17] formulated the inventory system of a single-product three-level multiperiod SC with uncertain demands and lead times as a robust mixed-integer linear program with minimized expected cost and total cost variation to determine the optimal s, S values of the inventory parameters. Using the proportional control approach, Wang and Disney [18] mitigated the amplification of order and inventory fluctuations in a state-space SC model with stochastic lead time.
The SC system is only considered as a linear system whether with the controllable lead time or with the uncontrollable lead time. However, it is worth noting that the SC system is dynamic in the operational process due to the influences of the uncertain customers’ demands and lead times. In this perspective, this leads to multiple possible strategies in manufacturing, delivering, and ordering products measured by the relation between upstream company’s inventory level and downstream company’s demand state. That is to say, the node companies of manufacturing or ordering can implement multiple strategies instead of one in different scenarios. In such a situation, a linear switching system with many modes can be performed instead of a unique SC model as well. Therefore, the SC system is nonlinear dynamic with piecewise linear characteristics. Nevertheless, the SC as a nonlinear dynamic system has been rarely addressed in the related literature.
Robust fuzzy control strategies for controlling nonlinear dynamic systems have been addressed broadly. For the nonlinear systems with uncertainties, Lee et al. [19] studied the fuzzy robust control problem for the continuous-time and discrete-time nonlinear systems with parametric uncertainties based on Takagi-Sugeno (T-S) fuzzy model and derived the sufficient conditions of robust stabilization in the sense of Lyapunov asymptotic stability; Yang and Zhao [20] presented a robust control approach for uncertain switched fuzzy system and designed a continuous state feedback controller to ensure the relevant closed-loop system is asymptotically stable for all allowable uncertainties. The nonlinear systems with time delay are also mentioned in a few literatures. Cui et al. [21] discussed the problem of robust H∞ control for a class of uncertain switched fuzzy time-delay systems described by T-S fuzzy model and derived a sufficient condition to guarantee the stability of the closed-loop systems. Further, Teng et al. [22] investigated the robust model predictive control of a class of nonlinear discrete system subjected to time delays and persistent disturbances. However, the robust control approaches in [19–22] cause higher conservative to guarantee the stability of the system by finding the common positive definite matrices.
In this paper, we will propose a fuzzy robust H∞ control strategy to restrain the impacts of lead times, switching actions among subsystems, and customers’ stochastic demands on the nonlinear dynamic SC system. By utilizing the concept of maximal overlapped-rules group (MORG), the control strategy can be obtained from T-S fuzzy system associated with robust H∞ control method, which can guarantee the stability of the system if the local common positive definite matrices in each MORG can be found. Therefore, the proposed control strategy can (i) reduce the conservatism as compared with the existing control approaches; (ii) make SC system robustly asymptotically stable; and (iii) realize soft switching among subsystems of the nonlinear dynamic SC. We make some comparisons with the robust H∞ control strategy to demonstrate the effectiveness of our proposed strategy.
The rest of this paper is arranged as follows. The fuzzy model of the nonlinear dynamic SC system with lead times is formulated in Section 2. Then Section 3 proposes a new fuzzy robust H∞ control strategy. Finally, Section 4 provides an illustrative example of a two-stage nonlinear SC system with the production lead time and the ordering lead time to verify the advantage of the proposed control strategy. Our conclusions are presented in Section 5.
2. Model Construction and Preliminaries2.1. Nonlinear Dynamic SC Fuzzy System
The formulation of a basic model of two-stage SC system with lead times (i.e., production lead time and ordering lead time) can be illustrated in Figure 1.
Basic dynamic model of two-stage SC system.
In Figure 1, xa(k) and xb(k) denote manufacturer a′ inventory level and retailer b′ inventory level at period k, respectively, ua(k) and ua(k-τa) denote the productions manufactured by manufacturer (a) at period k and with the production lead time τa, respectively, ub(k) and ub(k-τb) are the numbers of products ordered by retailer (b) at period k and with the ordering lead time τb, respectively, and wb(k) is the customers’ demands at period k.
Remark 1.
Figure 1 can describe 4 kinds of SC systems: (1) when a=1 and b=1, Figure 1 denotes the chain-type SC system; (2) when a=1 and b=2,3,…,t, Figure 1 denotes the distribution-type SC system; (3) when a=2,3,…,s and b=1, Figure 1 denotes retailers-centered multistage SC system, like supermarket. (4) when a=1,2,3,…,s and b=1,2,3,…,t, Figure 1 denotes the SC network system.
The basic dynamic mathematical model of the SC system is presented as follows:(1)xak+1=xak+uak+uak-τa-ubk,xbk+1=xbk+ubk+ubk-τb-wbk.
In the operational process of the SC system, the node companies will adopt different production or ordering strategies according to their own different inventory levels, which results in many different basic dynamic models, and the basic dynamic models can be called subsystems. Moreover, to reduce the total cost of this SC system, there exist switching actions among subsystems at different period k. Therefore, the dynamic SC system is a piecewise linear system, which can be also called a nonlinear system.
By utilizing the matrix theory and considering the total cost of the nonlinear SC system, the ith subsystem of (1) can be described as follows:(2)xk+1=Aixk+Biuk+∑e=1nBieuk-τe+Bwiwk,zk=Cixk+Diuk+∑e=1nDieuk-τe,where the subscript i corresponds to the SC being in the ith mode, x(k)=x1(k)x2(k)⋯xn(k)T (n=s+t) is the inventory state variable, u(k)=u1(k)u2(k)⋯un(k)T is the control variable, u(k-τe)=u1(k-τ1)u2(k-τ2)⋯un(k-τn)T is the control variable with lead times, w(k) is the customers’ demands variable, z(k) is the total cost of the SC system, Ai is coefficients matrix of the inventory strategy implementation, Bi is coefficients matrix of the productivity and ordering placement, Bie is coefficients matrix of the productivity and ordering placement during lead times, Bwi is coefficients matrix of customers’ demands, Ci is coefficients matrix of the inventory cost, Di is coefficients matrix of the manufacturing and ordering cost, and Die is coefficients matrix of the manufacturing and ordering cost with lead times.
This nonlinear SC system (2) is described with deviation values which are the differences between the actual values and the nominal values.
T-S fuzzy system is a powerful tool for processing nonlinear systems [23]. T-S fuzzy system consists of fuzzy rules that express local linear relationship between inputs and outputs of a system. Hence, based on T-S fuzzy system, we will establish a nonlinear SC fuzzy system.
For the nonlinear SC system (2), the ith fuzzy control rule can be described as follows:
Ri: if x1k is M1i, and,…, and xj(k) is Mji,…, and xnk is Mni, then(3)xk+1=Aixk+Biuk+∑e=1nBieuk-τe+Bwiwk,zk=Cixk+Diuk+∑e=1nDieuk-τe,xk=φk,k=0,1,…,N,where Ri (i=1,2,…,r) is the ith fuzzy rule, r is the number of if-then rules, Mji (j=1,2,…,n) is the fuzzy set of the inventory level, and φ(k) is the initial condition.
By singleton fuzzification, product inference, and center-average defuzzification, (3) can be inferred as (4)xk+1=∑i=1rhixkAixk+Biuk+∑e=1nBieuk-τe+Bwiwk,zk=∑i=1rhixkCixk+Diuk+∑e=1nDieuk-τe,where the membership function hixk=μi(x(k))/∑i=1rμi(x(k)), in which μi(x(k))=∏j=1nMji(xj(k)). Mji(xj(k)) represents the grade of membership of xj(k) in Mji. For all k, hi(x(k))≥0 and ∑i=1rhi(x(k))=1. For simplicity, we omit x(k) in hi(x(k)).
2.2. Fuzzy Control Strategy
To restrain the impacts caused by lead times, switching actions among subsystems, and customers’ stochastic demands on the nonlinear SC system, this paper will design a fuzzy robust H∞ control strategy. This strategy can incorporate the membership function of T-S fuzzy system into the robust H∞ control method to realize soft switching and make the system robustly asymptotically stable.
Based on the parallel distributed compensation scheme, the control law of the nonlinear SC system is formulated as follows.
Controller rule Ki is as follows: if x1(k) is M1i and,…, and xj(k) is Mji,…, and xn(k) is Mni, then(5)uk=-∑i=1rhiKixk,uk-τe=-∑i=1rhiKiexk-τe,where Ki and Kie denote the inventory feedback gains matrices of the ith local model. Ki is the coefficients matrix of production plan and ordering delivery; Kie is the coefficients matrix of production plan and ordering delivery during lead time.
Using the fuzzy controller (5), this paper intends to make the following system robustly asymptotically stable during lead times:(6)xk+1=∑i=1r∑j=1rhihjAi-BiKjxk-∑e=1nBieKjexk-τe+Bwiwk,zk=∑i=1r∑j=1rhihjCi-DiKjxk-∑e=1nDieKjexk-τe.
The inhibitory degree of this controller (5) can be described as the parameter γ; namely,(7)total cost2customers’ demands2≤γ,where ·2 is L2 norm [24]. The smaller the parameter γ is, the better the performance of this SC fuzzy system (6) is.
2.3. Preliminaries
Before proceeding, we will introduce our theorem by recalling the following definitions, proposition, and lemmas.
Definition 2 (see [<xref ref-type="bibr" rid="B25">25</xref>]).
For a given scalar γ>0 which denotes the disturbance attenuation level for a system, the nonlinear SC fuzzy system (6) is said to be robustly asymptotically stable with the γ constraint under the H∞ norm if two conditions as below are satisfied.
(1) The nonlinear SC fuzzy system (6) is robustly asymptotically stable when w(k)≡0.
(2) Under zero-initial condition, the total cost z(k) of the nonlinear SC fuzzy system (6) satisfies z(k)22<γw(k)22 for any nonzero w(k)∈l20,∞ and all admissible uncertainties.
Accordingly (5) is called a γ-suboptimal robust H∞ control law of the SC fuzzy system (6).
Definition 3 (see [<xref ref-type="bibr" rid="B26">26</xref>]).
A cluster of fuzzy sets {Fju,u=1,2,…,qj} are said to be a standard fuzzy partition (SFP) in the universe X if each Fju is a normal fuzzy set and Fjuu=1,2,…,qj are full-overlapped in the universe X. qj is said to be the number of fuzzy partitions of the jth input variable on X.
Definition 4 (see [<xref ref-type="bibr" rid="B26">26</xref>]).
For a given fuzzy system, an overlapped-rules group with the largest amount of rules is said to be a maximal overlapped-rules group (MORG).
Proposition 5 (see [<xref ref-type="bibr" rid="B26">26</xref>]).
If the input variables of a fuzzy system adopt SFPs, then all the rules in an overlapped-rules group must be included in a MORG.
Lemma 6 (see [<xref ref-type="bibr" rid="B27">27</xref>]).
For any real matrices Xi, Yi for 1≤i≤n, and S>0 with appropriate dimensions, we have(8)2∑i=1n∑j=1n∑k=1n∑l=1nhihjhkhlXijTSYkl≤∑i=1n∑j=1nhihjXijTSXij+YijTSYij,where hi1≤i≤n are defined as hiM(k)≥0, ∑i=1nhiM(k)=1.
Lemma 7.
For any real matrices Xij1≤i,j≤n, and S>0 with appropriate dimensions, the following inequality holds:(9)∑i=1n∑j=1n∑k=1n∑l=1nhihjhkhlXijTSXkl≤∑i=1n∑j=1nhihjXijTSXij.
Proof.
For Lemma 6, let X=Y; then we have(10)2∑i=1n∑j=1n∑k=1n∑l=1nhihjhkhlXijTSXkl≤∑i=1n∑j=1nhihjXijTSXij+XijTSXij=2∑i=1n∑j=1nhihjXijTSXij.Therefore, ∑i=1n∑j=1n∑k=1n∑l=1nhihjhkhlXijTSXkl≤∑i=1n∑j=1nhihjXijTSXij can be obtained.
3. Fuzzy Robust <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M139"><mml:mrow><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>∞</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> Control of Nonlinear SC
A fuzzy robust H∞ output-feedback controller for the T-S fuzzy system with uncertainties was recently introduced by [28]. It also came to use in [29] to restraint of the bullwhip effect for uncertain closed-loop SC system.
In this section, we also apply this idea of the fuzzy controller for the nonlinear SC fuzzy system (6) with lead times.
Theorem 8.
For a given scalar γ>0, if there exist local common positive definite matrices Pc and Qec in Gc such that the following linear matrix inequalities (LMIs) (11) and (12) are satisfied, then the supply chain fuzzy system (6) with lead times and SFP inputs is robustly asymptotically stable and the H∞ norm is less than a given bound γ:(11)-P̿∗∗M-ii-Pc-1∗N-ii0-I<0,i∈Ic,(12)-4P̿∗∗2M̿ij-Pc-1∗2N--ij0-I<0,i<j,i,j∈Ic,where P̿=Pc-∑e=1nQec∗∗0Q^∗00γ2I, Q^=diagQ1c⋯Qec⋯Qnc, M-ij=Mij-Bi1Kj1c⋯-BieKjec⋯-BinKjncBwi, Mij=Ai-BiKjc, N-ij=Nij-Di1Kj1c⋯-DieKjec⋯-DinKjnc0, Nij=Ci-DiKjc, M̿ij=(M-ij+M-ji)/2, N̿ij=(N-ij+N-ji)/2, 0 denotes the zero matrix, I denotes the identity matrix, Ic is a set of the rule numbers included in Gc, Gc denotes the cth MORG, c=1,2,…,∏j=1nmj-1, and mj is the number of the fuzzy partitions of the jth input variable.
Proof.
Consider two scenarios: first, if state input variables xk and xk+1 are in the same overlapped-rules group, the fuzzy system (6) will be proved to be robustly asymptotically stable. Then if state input variables xk and xk+1 are in the different overlapped-rules groups, the same result will be obtained.
Assume that the fuzzy system (6) contains f overlapped-rules groups; vdd=1,2,…,f is the operating region of the dth overlapped-rules group and Ld={the rule numbers included in the dth overlapped-rulesgroup}.
In the first scenario, the local model of the dth overlapped-rules group can be described as(13)xk+1=∑i∈Ld∑j∈LdhihjMijxk-∑e=1nBieKjecxk-τe+Bwiwk,zk=∑i∈Ld∑j∈LdhihjNijxk-∑e=1nDieKjecxk-τe,where Kjec is the state feedback control gain of production lead time and ordering lead time in the cth MORG.
Equation (13) can be represented as follows:(14)xk+1=∑i∈Ld∑j∈LdhihjM-ijx-k,zk=∑i∈Ld∑j∈LdhihjN-ijx-k,where(15)x-k=xkxk-τ1⋯xk-τe⋯xk-τnwkT.
Consider the discrete Lyapunov function:(16)Vdxk=xTkPcxk+∑e=1n∑ξ=k-τek-1xTξQecxξ.
And using Lemma 7 will supply(17)ΔVdxk=Vdxk+1-Vdxk=xTk+1Pcxk+1-xTkPcxk+∑e=1nxTkQecxk-xTk-τeQecxk-τe=∑i∈Ld∑j∈Ldhihj∑p∈Ld∑q∈Ldhphqx-TkM-ijTPcM-pqx-k-xTkPcxk+∑e=1nxTkQecxk-xTk-τeQecxk-τe=∑i∈Ld∑j∈Ldhihj∑p∈Ld∑q∈Ldhphqx-TkM-ijTPcM-pq-P-x-k=∑i∈Ld∑j∈Ldhihj∑p∈Ld∑q∈Ldhphqx-TkM-ij+M-ji2TPcM-pq+M-qp2-P-x-k=∑i∈Ld∑j∈Ldhihj∑p∈Ld∑q∈Ldhphqx-TkM̿ijTPcM̿pq-P-x-k≤∑i∈Ld∑j∈Ldhihjx-TkM̿ijTPcM̿ij-P-x-k,where P-=Pc-∑e=1nQec∗∗0Q^∗000 and M̿pq=(M-pq+M-qp)/2.
Thus ΔVdxk satisfies the relation(18)ΔVdxk≤∑i=j,i∈Ldhi2x-TkM-iiTPcM-ii-P-x-k+2∑i<ji∈Ld,j∈Ldhihjx-TkM̿ijTPcM̿ij-P-x-k.
Assume the customers’ demands wk≠0; H∞ performance index function J1 can be presented by(19)J1=∑k=0N-1zTkzk-γ2wTkwk.
The above equation can be rewritten as(20)J1=∑k=0N-1zTkzk-γ2wTkwk+ΔVdxk-VdxN≤∑k=0N-1zTkzk-γ2wTkwk+ΔVdxk.Substituting (18) into (20), we have(21)J1≤∑k=0N-1∑i=j,i∈Ldhi2x-TkM-iiTPcM-ii-P̿+N-iiTN-iix-k+2∑k=0N-1∑i<j,i∈Ldj∈Ldhihjx-TkM̿ijTPcM̿ij-P̿+N̿ijTN̿ijx-k.
Applying Schur complement to (11) and (12) results in M-iiTPcM-ii-P̿+N-iiTN-ii<0 and M̿ijTPcM̿ij-P̿+N̿ijTN̿ij<0. Then, J1<0 can be obtained; that is, zTkzk<γ2wTkwk; moreover, let N→+∞; then we have z(k)22<γ2w(k)22. As a result, the SC system (14) is proved to be asymptotically stable in the case of w(k)≠0.
On the other hand, if w(k)≡0, it is obvious that (18) is equivalent to the following inequality:(22)ΔVdxk≤∑i=j,i∈Ldhi2x-TkM-iiTPcM-ii-P̿x-k+2∑i<ji∈Ld,j∈Ldhihjx-TkM̿ijTPcM̿ij-P̿x-k.
According to (11) and (12) we can obtain M-iiTPcM-ii-P̿<0 and M̿ijTPcM̿ij-P̿<0, respectively. Accordingly, we can conclude that ΔVd(x(k))<0. As a result, the state feedback controller can ensure the local system (6) robustly asymptotically stable in the dth overlapped-rules group.
In the second scenario: a characteristic function in any overlapped-rules group is constructed as follows:(23)λd=1,xk∈vd0,xk∉vd,where ∑d=1fλd=1; then the global model of the discrete fuzzy system in the input universe of the discourse can be described as follows:(24)xk+1=∑d=1fλd∑i∈Ld∑j∈LdhihjM-ijx-k,zk=∑d=1fλd∑i∈Ld∑j∈LdhihjN-ijx-k.
In the following, based on the definitions of Pm=∑d=1fλdPc and Qem=∑d=1fλdQec, a piecewise Lyapunov function can be introduced in the input universe of the discourse as follows:(25)Vxk=xTkPmxk+∑e=1n∑ξ=k-τek-1xTξQemxξ=xTk∑d=1fλdPcxk+∑e=1n∑ξ=k-τek-1xTξ∑d=1fλdQecxξ=∑d=1fλdxTkPcxk+∑e=1n∑ξ=k-τek-1xTξQecxξ=∑d=1fλdVdxk.
We first assume that the customers’ demands wk≠0. For (24) we can obtain J2=∑k=0N-1∑d=1fλd[zT(k)z(k)-γ2wT(k)w(k)] after the H∞ performance index function J1=∑k=0N-1[zT(k)z(k)-γ2wT(k)w(k)] is considered. J2<0 can be obtained through a similar procedure; that is, zT(k)z(k)<γ2wT(k)w(k); moreover, let N→+∞; then we have z(k)22<γ2w(k)22. As a result, (24) is proved to be asymptotically stable in the case of w(k)≠0.
On the other hand, if w(k)≡0, we have(26)ΔVxk=Vxk+1-Vxk=∑d=1fλdVdxk+1-∑d=1fλdVdxk=∑d=1fλdVdxk+1-Vdxk=∑d=1fλdΔVdxk<0.
Hence, in any overlapped-rules group, (24) with w(k)≡0 is asymptotically stable by the fuzzy controller (5).
Therefore, according to Proposition 5, we can conclude that the fuzzy system (6) is robustly asymptotically stable with Condition (11) and Condition (12) by resorting to find local common positive definite matrices Pc and Qec in Gc. This completes the proof of the theorem.
In Theorem 8, for a given H∞ performance index γ, we can obtain the robust stabilization conditions of (6), which are represented by a set of matrix inequalities in (11) and (12). Subsequently we will show that such inequalities can be transformed into LMIs when designing the actual H∞ controllers. Note that the feasibility of LMIs can be easily achieved by using the LMI toolbox in MATLAB.
Theorem 8 can be recast as the LMI problem by the following Theorem 9.
Theorem 9.
For the supply chain fuzzy system (6) with lead times and SFP inputs, if there exist a given scalar γ>0, local common positive definite matrices Pc and Qec, and matrices Kic, Kjc, Kiec, Kjec in Gc, such that the following LMIs are satisfied, then the supply chain fuzzy system (6) is robustly asymptotically stable under the performance γ:(27)-Pc+∑e=1nQec∗∗∗∗0-Q^∗∗∗00-γ2I∗∗Ai-BiKic-Π1Bwi-Pc∗Ci-DiKic-Π200-I<0,i∈Ic,-4Pc+4∑e=1nQec∗∗∗∗0-4Q^∗∗∗00-4γ2I∗∗Ai-BiKjc+Aj-BjKic-Φ1Bwi+Bwj-Pc∗Ci-DiKjc+Cj-DjKic-Φ200-I<0,i<j,i,j∈Ic,where(28)Q^=diagQ1c⋯Qec⋯Qnc,Π1=Bi1Ki1c⋯BieKiec⋯BinKinc,Π2=Di1Ki1c⋯DieKiec⋯DinKinc,Φ1=Bi1Kj1c+Bj1Ki1c⋯BieKjec+BjeKiec⋯BinKjnc+BjnKinc,Φ2=Di1Kj1c+Dj1Ki1c⋯DieKjec+DjeKiec⋯DinKjnc+DjnKinc,Ic is a set of the rule numbers included in Gc, Gc denotes the cth MORG, c=1,2,…, ∏j=1nmj-1, and mj is the number of the fuzzy partitions of the jth input variable.
Proof.
The proof is analogous to that of Theorem 8. For simplicity, the similar sections are truncated. Theorem 9 can be easily demonstrated by using matrix transformations and the Schur complement. The main procedure is as follows.
In order to solve the LMIs, (11) can be expressed further as follows:(29)-Pc+∑e=1nQec∗∗∗∗0-Q^∗∗∗00-γ2I∗∗Ai-BiKic-Π1Bwi-Pc-1∗Ci-DiKic-Π200-I<0.
Taking the congruence transformation with diagI,I,I,Pc,I easily verifies(30)-Pc+∑e=1nQec∗∗∗∗0-Q^∗∗∗00-γ2I∗∗Ai-BiKic-Π1Bwi-Pc∗Ci-DiKic-Π200-I<0.
Clearly, (12) is equivalent to(31)4-Pc+∑e=1nQec∗∗∗∗0-4Q^∗∗∗00-4γ2I∗∗Ai-BiKjc+Aj-BjKic-Φ1Bwi+Bwj-Pc-1∗Ci-DiKjc+Cj-DjKic-Φ200-I<0.
Taking the congruence transformation with diagI,I,I,Pc,I results in(32)-4Pc+4∑e=1nQec∗∗∗∗0-4Q^∗∗∗00-4γ2I∗∗Ai-BiKjc+Aj-BjKic-Φ1Bwi+Bwj-Pc∗Ci-DiKjc+Cj-DjKic-Φ200-I<0.
Remark 10.
(1) Compared with the common robust control approach, the proposed fuzzy robust control strategy is less sensitive to the variations of system parameters and can realize soft switching and make the system robustly asymptotically stable. (2) The proposed fuzzy robust control strategy requires only finding local common positive definite matrices in each MORG to check the robust stability of T-S fuzzy system. Therefore, the new control strategy can reduce the conservation and difficulty of the common Lyapunov function approach. (3) The performance and validity of the controller (5) depend on the parameter γ, which will play an important role in the existence of some matrices satisfying a series of LMIs. The smaller the parameter γ is, the better the performance of the system will be. Therefore, when γ is set too small in order to obtain the better performance of the system, some matrices satisfying a series of LMIs may not exist. For a smaller γ, if some matrices satisfying a series of LMIs do not exist, γ will be constantly set to a bigger value little by little until some matrices satisfying a series of LMIs exist.
Remark 11.
The numbers of LMIs satisfied to check the stability of a system for Theorem 9 in this paper and for Theorem 3 in [30] are ∏j=1nmj-12(n-1)(2n+1)+6∏j=1nmj-1 and (1/2)∏j=1nmj3+(1/2)∏j=1nmj2+2∏j=1nmj+5, respectively.
Table 1 shows the comparison results of the numbers of LMIs satisfied between Theorem 3 in [30] and Theorem 9 in this paper, where n denotes the number of state variables and υ and ϑ denote the numbers of LMIs satisfied for Theorem 3 in [30] and Theorem 9 in this paper, respectively.
The comparison results of the numbers of LMIs satisfied between Theorem 3 in [30] and Theorem 9 in this paper.
n
m
r
υ
ϑ
1
3
3
29
18
1
4
4
53
27
2
3
9
428
64
2
4
16
2213
144
From Table 1, we know that the difference of υ and ϑ becomes more and more large along with the increase of r.
4. Simulation Research4.1. Modeling of Two-Stage Chain-Type SC Fuzzy System
We illustrate the effectiveness of the fuzzy robust H∞ control strategy described in Section 3, which is based on Chinese compressors manufacturing industry. This industry can be considered as a two-stage chain-type nonlinear SC system with production lead time and ordering lead time.
The manufacturer’s production strategies are set as follows. (1) When the manufacturer’s inventory level x1(k)<0, the manufacturer manufactures the products according to the JIT (Just In Time) mode in order to meet the retailer’s order needs. (2) When x1(k)∈0,I0m (I0m is the manufacturer’s expected inventory), the manufacturer normally manufactures the products to satisfy the retailer’s demand.
The retailer’s order strategies are set as follows. (1) When the retailer’s inventory level x2(k)<0, the retailer asks for help from other SC. (2) When x2(k)∈0,I0r (I0r is the retailer’s expected inventory), the retailer normally orders the products from the manufacturer.
As shown in Figure 2, the fuzzy partitions of x1(k) and x2(k) are H1t(x1(k)) (t=1,2) and H2s(x2(k))(s=1,2), respectively, and meet the conditions of SFP. Set M11=M12=H11, M13=M14=H12, M21=M23=H21, and M22=M24=H22. In Figure 2, I0m=0.1 (×106 sets), I0r=0.8 (×105 sets), and Immax and Irmax denote the manufacturer’s maximal inventory level and the retailer’s maximal inventory level, respectively.
Fuzzy membership functions of inventory input variables.
In Figure 2, there is one MORG named S that includes 4 rules in this system.
We can obtain this SC system with the production lead time and the ordering lead time as follows.
Ri: if x1(k) is M1i and x2(k) is M2i, then(33)xk+1=∑i=1rhiAixk+Biuk+Bi1uk-τ1+Bi2uk-τ2+Bwiwk,zk=∑i=1rhiCixk+Diuk+Di1uk-τ1+Di2uk-τ2,where r=4, x(k)=x1(k)x2(k)T, u(k)=u1(k)u2(k)T, τ1 is the production lead time, τ2 is the ordering lead time, w(k)=0d(k)T, d(k) denotes the customers’ demands, and the system parameters are set as follows: A1=0, A2=0001, A3=1000, A4=1001, B1=B3=1-101+λ, B2=B4=1-101, B11=B21=B31=B41=1000, B12=B22=B32=B42=0001, Bw1=Bw2=Bw3=Bw4=000-1, C1=0, C2=0cr2, C3=cr10, C4=cr1cr2, D1=cmJc0+c0L, D2=cmJ0, D3=cmc0+c0L, D4=cmc0, D11=D21=cmJ0, D31=D41=cm0, D12=D32=0c0+c0L, D22=D42=0c0, where cr1 and cr2 are the manufacturer’s unit inventory cost and the retailer’s unit inventory cost, respectively, cr1=0.13 (×106 RMB) and cr2=0.30 (×106 RMB), cm is the unit manufacturing cost when x1(k)∈0,D0m, cm=1.06 (×106 RMB), cmJ is the unit manufacturing cost under JIT condition, cmJ=1.87 (×106 RMB), c0 is the retailer’s unit ordering cost when x2(k)∈0,D0r, c0=1.66 (×106 RMB), c0L is the retailer’s unit ordering cost from the manufacturers in the different SCs, c0L=1.91 (×106 RMB), and λ is the supplying rate from other SCs, 0<λ≤1, λ=0.52.
Then the fuzzy controller is designed as follows:(34)Ki: Ifx1kisM1i,x2kisM2i,thenuk=-∑i=1rhiKi1xk,uk-τ1=-∑i=1rhiKi11xk-τ1,uk-τ2=-∑i=1rhiKi21xk-τ2.
4.2. Simulation Analysis
When γ=0.6, this SC fuzzy system is robust stable by being found the following local matrices in MORG:(35)P1=292.5161-55.5053-55.5053119.3743,Q11=Q21=55.8514-17.0739-17.073911.1860.
In verifying the advantages of the proposed fuzzy robust H∞ control strategy for this nonlinear SC fuzzy system, the simulation experiments will be performed.
The initial values of states are x10=0.9 (×105 sets), x20=0.1 (×105 sets), and the normal values are set as x→1(k)=1.5 (×106 sets), x→2(k)=0.5 (×105 sets), u→1(k)=0.1(×106 sets), u→2(k)=0.6 (×105 sets), z→(k)=0.5 (×106 RMB), the production lead time τ1=2 (in weeks) and the ordering lead time τ2=2 (in weeks), and the customers’ demands d(k)~N(3,0.52). The simulation results are expressed as the actual values.
To show the superiority of our proposed control strategy, the robust H∞ control strategy and our control strategy are provided for comparison. The robust H∞ control strategy can be described as follows:(36)Ki: Ifx1kisM1i,x2kisM2i,thenuk=-Ki1xk,uk-τ1=-Ki11xk-τ1,uk-τ2=-Ki21xk-τ2.
Figures 3, 5, and 7 illustrate the evolution processes of inventory levels, production and ordering quantity, and total cost under the robust H∞ control strategy.
Evolution processes of inventory levels under the robust H∞ control strategy.
By using our proposed control strategy, that is to say, fuzzy robust H∞ control strategy, the simulation results are shown in Figures 4, 6, and 8.
Evolution processes of inventory level under the fuzzy robust H∞ control strategy.
Evolution processes of production and ordering quantity under the robust H∞ control strategy.
Evolution processes of production and ordering quantity under the fuzzy robust H∞ control strategy.
Evolution process of total cost under the robust H∞ control strategy.
Evolution process of total cost under the fuzzy robust H∞ control strategy.
Figures 3 and 4 are selected as an example to analyze the control effects of two control approaches. As shown in Figures 3 and 4, the differences of the crests and the troughs for the manufacturer’s inventory level x1(k) under the robust H∞ control strategy and the fuzzy robust H∞ control strategy are 30×104 sets-15 ×104sets=15×104 sets and 15×104 sets-12×104sets=2×104 sets, respectively.
5. Conclusion
The robust stabilization problem of the operational process of the dynamic SC system with lead times has been studied. Utilizing T-S fuzzy system, we establish the nonlinear dynamic SC fuzzy system. To restrain the disturbances of lead times, switching actions among subsystems, and customers’ stochastic demands, a new fuzzy robust H∞ control strategy has been proposed by utilizing the definition of MORG. In addition, this strategy can guarantee that the nonlinear SC system with lead times is robustly asymptotically stable. In the simulation research, we have discussed the operational process of two-stage nonlinear SC system with the production lead time and the ordering lead time, and the simulation tests have verified the advantage of the proposed fuzzy robust control strategy. Our designed strategy can be applied to some node companies in supply chain system.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Social Science Foundation of China (17BGL222).
ZimmerK.Supply chain coordination with uncertain just-in-time deliveryZhangG.ShangJ.LiW.Collaborative production planning of supply chain under price and demand uncertaintyParkS.LeeT.-E.SungC. S.A three-level supply chain network design model with risk-pooling and lead timesTersineR. J.HummingbirdE. A.Lead-time reduction: the search for competitive advantageMahajanS.VenugopalV.Value of information sharing and lead time reduction in a supply chain with autocorrelated demandLengM.ParlarM.Lead-time reduction in a two-level supply chain: non-cooperative equilibria vs. coordination with a profit-sharing contractLiY.XuX. J.YeF.Supply chain coordination model with controllable lead time and service level constraintGlockC. H.Lead time reduction strategies in a single-vendorsingle-buyer integrated inventory model with lot size-dependent lead times and stochastic demandJhaJ. K.ShankerK.An integrated inventory problem with transportation in a divergent supply chain under service level constraintHeydariJ.Lead time variation control using reliable shipment equipment: An incentive scheme for supply chain coordinationGarciaC. A.IbeasA.HerreraJ.VilanovaR.Inventory control for the supply chain: An adaptive control approach based on the identification of the lead-timeXuH.RongG.A minimum variance control theory perspective on supply chain lead time uncertaintyTaleizadehA. A.NiakiS. T. A.ShafiiN.MeibodiR. G.JabbarzadehA.A particle swarm optimization approach for constraint joint single buyer-single vendor inventory problem with changeable lead time and (r,Q) policy in supply chainLiC.LiuS.A robust optimization approach to reduce the bullwhip effect of supply chains with vendor order placement lead time delays in an uncertain environmentGarciaC. A.IbeasA.VilanovaR.HerreraJ.Lead-time identification for inventory control of the supply chainProceedings of the 20th Mediterranean Conference on Control & AutomationJuly 2012Barcelona, Spain728733HanX.-J.FengA.-M.ZhangB.-L.Approximate optimal inventory control of supply chain networks with lead timeProceedings of the 27th Chinese Control and Decision Conference, CCDC 2015May 20154523452810.1109/CCDC.2015.71627222-s2.0-84945584907MovahedK. K.ZhangZ.-H.Robust design of (s,S) inventory policy parameters in supply chains with demand and lead time uncertaintiesWangX.DisneyS. M.Mitigating variance amplification under stochastic lead-time: the proportional control approachLeeH. J.ParkJ. B.ChenG.Robust fuzzy control of nonlinear systems with parametric uncertaintiesYangH.ZhaoJ.Robust control for a class of uncertain switched fuzzy systemsCuiY.LiuK.ZhaoY.WangX.Robust H∞ control for a class of uncertain switched fuzzy time-delay systems based on T-S modelsTengL.WangY.CaiW.LiH.Robust model predictive control of discrete nonlinear systems with time delays and disturbances via T–S fuzzy approachRuiyunQi.GangT.JiangB.TanC.Adaptive control schemes for discrete-time T-S fuzzy systems with unknown parameters and actuator failuresProceedings of the 2011 American Control ConferenceJune 2011San Francisco, CA, USA3748375310.1109/ACC.2011.5990630LiC. M.TianX. M.Control based on LMIs for a class of time-delay switched systemLiuX. D.ZhangQ. L.Approaches to quadratic stability conditions and H∞ control designs for T-S fuzzy systemsXiuZ.-H.RenG.Stability analysis and systematic design of Takagi-Sugeno fuzzy control systemsGuanX. P.ChenC. L.Delay-dependent guaranteed cost control for T-S fuzzy systems with time delaysZhangS.HouY.ZhaoX.Robust stabilization for discrete uncertain Takagi-Sugeno fuzzy systems based on a piecewise Lyapunov functionZhangS.LiX.ZhangC.A fuzzy control model for restraint of bullwhip effect in uncertain closed-loop supply chain with hybrid recycling channelsZhangB.XuS.Delay-Dependent Robust H∞ control for uncertain discrete-time fuzzy systems with time-varying delays